Subcritical Instability Research Articles

Parametric resonance for pipes conveying fluid in thermal environment

Parametric resonance is a kind of specific fluid-induced vibration for pipes conveying fluid. This paper focuses on revealing the qualitative characteristics of parametric resonances of the pipe with pulsating fluid speed in thermal environment, and compares the differences between the resonance characteristics in subcritical and supercritical regions. According to the generalized Hamilton's principle, the partial-differential-integral governing equation of a straight pipe is established. Under simply-supported boundary conditions, the non-trivial equilibrium configuration is obtained analytically. The governing equation of a supercritical pipe is derived by coordinate substitution on the basis of the new equilibrium configuration. The approximate analytical solution of parametric resonance for a pipe conveying fluid in a thermal environment is obtained by using the direct multi-scale method. The approximate analytical solution is verified to be reliable by using the Runge-Kutta method. The stability bounds of the parameters inducing parametric resonance are investigated via the introduced analytical method. The results show that the influence of temperature increment on sub-harmonic resonance is non-monotonic. In the subcritical region, when the temperature increment increases, the unstable region widens. This means that the system is more prone to parametric resonance, and the response amplitude decreases. In the supercritical region, when the temperature increment increases, the unstable bandwidth decreases, and the response amplitude increases. When viscous damping is increased or the average velocity is decreased, the sub-critical instability region decreases, and the supercritical instability region increases. With the increase in the pulsation velocity, the unstable bandwidth of subcritical and supercritical regions will increase, and parametric resonance is more likely to occur.

Global stability of Bénard–Marangoni convection in an anisotropic porous medium

Surface tension is essential in many industrial applications, especially where the liquid surface is in contact with the environment, such as crystal growth, semiconductor manufacturing, and welding. The present article reports a numerical analysis of convection induced by the combined effects of buoyancy force and surface tension in an infinitely extended horizontal fluid-saturated anisotropic porous layer with high permeability. We assume that the bottom boundary is rigid and the top is exposed to the air. Biot numbers establish general thermal conditions at both ends instead of commonly used adiabatic and isothermal boundary conditions. The anisotropy of the porous structure results in thermal and mechanical anisotropy parameters while examining layer's stability. The Chebyshev Tau technique yields the critical Marangoni number, MaLc and MaEc, representing linear and energy stability boundaries. We compare the constraints obtained from linear and energy analyses and conclude that the energy bounds for the current problem are less than linear bounds, indicating subcritical instabilities may exist. It is also observed that thermal anisotropy and Biot numbers stabilize the system. In contrast, mechanical anisotropy and the Darcy number advance the onset of convection. The existing results of limiting cases of the present problem are recovered with remarkable accuracy.

Flow instabilities of second-grade fluid over a stretching sheet

The work at hand studies thin-film flow instabilities in a non-Newtonian second-grade fluid moving over a stretching sheet. The analysis considers the free surface evolution equation of the thin second-grade viscoelastic fluid model based on the long-wave approximation method. We undertook a linear stability analysis using a normal mode approach and obtained expressions for the linear time growth rate, linear phase speed, and critical wavenumber, which illustrates the linear stability structure of the flow system. Furthermore, the multi-scale analysis of the fluid’s weakly non-linear stability characteristics reveals the presence of subcritical instability in the linear stability zone. The study shows that the varying flow parameters affect the threshold amplitude, which influences the stable and unstable zones in the subcritical region. The outcomes show a stabilizing effect spanning the range of non-dimensional parameters, such as the second-grade parameter ranging from 0 to 100 in values, the surface tension parameter ranging from 0 to 100 in values, and the modified Froude number from 0 to 100 in values.

Hydrodynamics and instabilities of a falling liquid film with an insoluble surfactant

In this study, we investigate the linear and weakly nonlinear stability of a liquid film flowing down an inclined plane with an insoluble surfactant. First, the nonlinear evolution equations of a liquid film thickness and surfactant concentration are derived using the long-wave expansion method at a moderate Reynolds number (0 < Re ≤ 20). The linear stability of the flow is examined using the normal-mode method, and the linear stability criterion and critical Reynolds number Rec are obtained. The results reveal the destabilizing nature with increasing Reynolds number Re and the stabilizing nature with increasing Marangoni number M. Second, the nonlinear equations described by the complex Ginzburg–Landau equation are obtained using the multiple-scale method to investigate the weakly nonlinear stability of the system. The results show that a new linear instability region appears above the neutral stability curve caused by the solute-Marangoni effect, which develops into a supercritical stable zone under the influence of nonlinear factors. Increasing M generally improves the stability of the flow but continuing to increase M under the condition of M > Mc (critical Marangoni number) improves the nonlinear instability in the region and transforms part of the unconditional stability zone into a subcritical instability zone. The increase in Re extends an explosive unstable zone and reduces the threshold amplitude in the subcritical unstable zone. In contrast, the unconditional stable zone decreases and disappears after increasing Re to a specific value, which reflects the destabilizing effect of Re on the nonlinear zone of the flow.

Linear and nonlinear stability of Rayleigh–Bénard convection with zero-mean modulated heat flux

Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( ${Ra}_{cr}$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The ${Ra}_{cr}$ value depends on the non-dimensional frequency $\omega$ of the boundary heat-flux modulation. Floquet theory is used to find ${Ra}_{cr}$ for linear stability, and the energy method is used to find ${Ra}_{cr}$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $\omega$ , with only the latter at large $\omega$ . For a given frequency, the linear stability ${Ra}_{cr}$ is generally higher than the nonlinear stability ${Ra}_{cr}$ , as expected. For large $\omega$ , ${Ra}_{cr} \omega ^{-2}$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $\omega$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.

Analytical study of Brinkman–Bénard convection in a bidisperse porous medium: Linear and weakly nonlinear study

Linear and weakly nonlinear stability analyses of Brinkman–Bénard convection of a Newtonian fluid saturating a bidisperse porous medium (BDPM) are made. Local-thermal-non-equilibrium (LTNE) is assumed between the fluid and the porous spheres (micro-pores) that make up the macro porous medium. Two coupled linear momentum equations are considered one each for the macro- and micro-pores. Results of mono-disperse porous medium (MDPM) with solid spheres are recovered as a limiting case of the present study. Further, in the case of both types of porous media considered the results of Darcy–Bénard and Brinkman–Bénard convection are extracted under suitable limiting procedures. To keep the work analytical, we reduce the intractable hexa-modal Lorenz model with four quadratic nonlinearities into the tractable mono-modal Stuart–Landau equation (SLE) with cubic and quintic nonlinearities. Subcritical instability is discounted in the study thereby suggesting that cubic SLE and cubic–quintic SLE both expound similar results qualitatively. The concept of a BDPM is shown to be meaningful only when the pores are not large, and when they are very small, then the MDPM assumption applies. Similar observation can be made when the ratio of permeabilities is large. The presence of micro-pores does not alter the size of the convective cell significantly at the onset. The present study reiterates the findings of several earlier works.

Nonlinear instability analysis of internally heated convection in a porous layer with the impact of magnetic field and variable gravity

AbstractThe impact of the magnetic field, heat source, and variable gravity field on the stability of convective phenomena in a porous layer are investigated numerically. Three types of gravity variations, such as linear, parabolic, and cubic functions are considered. For linear and nonlinear theories, the method of normal modes has been employed to solve governing dimensionless equations. It is found that the onset of convection is delayed by increasing the Hartmann number and gravity variation parameter and enhancement of the internal heat source makes the system stable. The subcritical instability region increases as the Hartmann number and gravity variation parameter increase.

Thermorheological effect on Rayleigh–Bénard magnetoconvection in a biviscous Bingham fluid with rough boundary condition on velocity and Robin boundary condition on temperature

AbstractThe thermorheological effect on the onset of Rayleigh–Bénard convection in a biviscous Bingham fluid in the presence of a horizontal magnetic field is investigated considering rough boundary conditions on velocity and Robin boundary conditions on temperature. The viscosity of the electrically conducting fluid is assumed to be sensitive to temperature variation. Linear and global nonlinear stability analyses are performed using the Chebyshev pseudospectral method to determine the existence of instability or otherwise. A general interpretation is made from the results to show the effects of the magnetic field and the variable viscosity on the system's stability. The biviscous Bingham parameter and the Chandrasekhar number are shown to have a delay in the onset of convection, while the effect of temperature sensitivity is to advance the onset. It is found that the results of linear and global nonlinear stability are not in agreement, so the region of subcritical instability exists. Also, the results obtained for Rayleigh–Bénard convection agree pretty well with those of Platten and Legros and Siddheshwar et al. for the limiting cases.

LINEAR AND NONLINEAR CONVECTIVE INSTABILITIES OF A MAXWELL-CATTANEO BIDISPERSIVE LAYER IN LOCAL THERMAL NON-EQUILIBRIUM

The linear and nonlinear instabilities of a horizontal Nield-Kuznetsov bidispersive layer in local thermal non-equilibrium subject to the Maxwell-Cattaneo (MC) effect is studied. In the absence of the MC effect, we prove that only a steady mode can exist. The presence of the MC effect, which introduces a fundamental change to the heat equation, gives rise to oscillatory motions. The linear stability of the layer can then take the form of steady or oscillatory motion. The stability boundary shows that the oscillatory stability boundary bifurcates from that of the steady mode to develop a situation of lower energy for instability. The presence of microporosity tends to inhibit the instability of oscillatory motions and introduces a mean flux that is uniform across the layer. A formal nonlinear analysis leads to two coupled evolution equations of the Landau-Stuart form, for two linearly preferred oscillatory waves propagating in opposite directions. The equations reduce to one for the steady mode. It is shown that the linearly unstable modes can take one of four types of nonlinear growth: supercritical stability, nonlinear instability, subcritical instability, or stability, depending on the relative values of the medium parameters. Although the two amplitudes start to grow interactively, the supercritical solutions always occur in the form of one amplitude, while the other dies out. However, the solution is different from that obtained by considering each wave separately.

The effects of the Soret and slip boundary conditions on thermosolutal convection with a Navier–Stokes–Voigt fluid

In this paper, we study the problem of thermosolutal convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from above or below. This problem is studied under the effects of Soret and slip boundary conditions. Both linear and nonlinear stability analyses are employed. When the layer is heated from below and salted from above, the boundaries exhibit great concordance, resulting in a very narrow region of probable subcritical instabilities. This proves that linear analysis is reliable enough to forecast the beginning of convective motion. The Chebyshev collocation technique and QZ algorithm have been used to solve systems of linear and nonlinear theories. For thermal convection in a dissolved salt field with a complex viscoelastic fluid of the Navier–Stokes–Voigt type, instability boundaries are computed. When the convection is of the oscillatory type, the Kelvin–Voigt parameter is observed to play a crucial role in functioning as a stabilizing agent. This effect's quantitative size is shown.

A weakly nonlinear analysis of the effect of vertical throughflow on Darcy–Bénard convection

The aim of this paper is to analyze the effect of a downward vertical net mass flow on the type of instability that occurs in a horizontal fluid-saturated porous layer that is heated from below. The strength of the downflow is modelled by Pe, the Péclet number. First, we prove the validity of the principle of exchange of stabilities; consequently, we perform a linear instability analysis of the basic steady flow to determine the critical Darcy–Rayleigh number for the onset of steady convective instability as a function of Pe. Then, a weakly nonlinear stability analysis is performed to determine the smallest value of the Péclet number for which the onset of convection corresponds to a subcritical instability.

Nonlinear stability analysis of penetrative convection in ferrofluids via internal heating

The phenomenon of natural convection driven by internal heating (also known as penetrative convection) has widespread applications in several branches of engineering. The present article investigates the onset of penetrative convection in a horizontal layer of ferrofluids. We assume four different supply functions generate uniform/non-uniform internal heating in the layer. Two sets of realistic boundary conditions (rigid–rigid and rigid-free) are chosen for analysis instead of commonly used artificial boundary conditions of free–free boundaries. The Rayleigh number serves as the eigenvalue of the system of governing equations and is obtained using linear and energy stability analyses. The Chebyshev Tau method is employed to numerically solve the eigenvalue problems obtained through both analyses. It was observed that the energy bounds are lower than the linear ones, indicating the existence of subcritical instabilities for the present problem. The quantitative results obtained for both linear and nonlinear analyses reveal that the Rayleigh number for rigid-free boundaries is always less than that for rigid–rigid boundaries. Moreover, it was also observed that the magnetic parameters advance the onset of penetrative convection.

Nonlinear magneto convection in an inclined porous layer with artificial neural network prediction

The onset of magnetoconvection in an inclined porous layer is investigated. The effects of physical parameters, such as the Rayleigh number, the inclination angle, and the Hartmann number, are examined. The system becomes more stable by increasing the inclination angle and Hartmann number. It is noted that the transverse rolls are more stable than the longitudinal rolls. A comparison between the linear and nonlinear instability analysis is discussed in more detail. The threshold values for the longitudinal and transverse rolls provide the subcritical instability region. It is noted that the subcritical instability region increases as the inclination angle also increases, whereas it remains unchanged as the Hartmann number increases. Besides, an artificial neural network (ANN) model using the Levenberg‐Marquardt backpropagation algorithm is employed to predict the distribution of critical Rayleigh numbers for both linear and nonlinear stability analyses. The optimal number of neurons in the hidden layer is selected on the basis of coefficient of determination, root mean square error, and root mean relative error. The simulated critical Rayleigh numbers obtained by the numerical study and the predicted critical Rayleigh numbers by the ANN coincides.

Viscoelasticity-Induced Instability in Plane Couette Flow at Very Low Reynolds Number

Elasto-inertial turbulence (EIT), a new turbulent state found in polymer solutions with viscoelastic properties, is associated with drag-reduced turbulence. However, the relationship between EIT and drag-reduced turbulence is not currently well-understood, and it is important to elucidate the mechanism of the transition to EIT. The instability of viscoelastic fluids has been studied in a canonical wall-bounded shear flow to investigate the transition process of EIT. In this study, we numerically deduced that an instability occurs in the linearly stable viscoelastic plane Couette flow for lower Reynolds numbers, at which a non-linear unstable solution exists. Under instability, the flow structure is elongated in the spanwise direction and regularly arranged in the streamwise direction, which is a characteristic structure of EIT. The regularity of the flow structure depends on the Weissenberg number, which represents the strength of elasticity; the structure becomes disordered under high Weissenberg numbers. In the energy spectrum of velocity fluctuations, a steep decay law of the structure’s scale towards a small scale is observed, and this can be recognized as a ubiquitous feature of EIT. The existence of instability in viscoelastic plane Couette flow supports the idea that the transitional path toward EIT may be mediated by subcritical instability.

Stability Analysis of Thin Power-Law Fluid Film Flowing down a Moving Plane in a Vertical Direction

This paper analyses the stability of thin power-law fluid flowing down a moving plane in a vertical direction by using the long-wave perturbation method. Linear and nonlinear stability are discussed. The linear stable region increases as the downward speed increases and the power-law index increases. More accurate results are obtained on the coefficients of the nonlinear generalized kinematic equation in the power-law part. The regions of sub-critical instability and absolute stability are expanded when the downward movement of plane increases, or the power-law index increases, and meanwhile the parts of supercritical stability and explosive supercritical instability are compressed. By substituting the power-law index and moving speed into the generalized nonlinear kinematic equation of the power-law film on the free surface, the results can be applied to estimate the stability of the thin film flow in the field of engineering.

Translation-deformation coupling effects on the Rayleigh instability of an electrodynamically levitated charged droplet.

The breakup pathway of the Rayleigh fission process observed in the past experiments carried out using high-speed imaging of a charged drop levitated in an AC quadrupole trap has shown to exhibit several cycles of shape and center-of-mass oscillations followed by asymmetric breakup by ejecting a jet in the upward direction (i.e., opposite to the direction of gravity). We recently attempted to explain this using boundary integral simulations in the Stokes flow limit, wherein the position of the droplet and the polarity of the end cap electrodes were assigned using physical arguments, and the center-of-mass motion was not estimated consistently invoking quasi-static conditions. In this work, we explain the experimental observation of upward breakup of charged droplets in a quadrupolar field, using numerical calculations based on the boundary element method considering inviscid droplets levitated electrodynamically using quadrupole electric fields. The center-of-mass motion and the end cap are consistently calculated in the numerical scheme. The simulations show that the gravity-induced downward shift in the equilibrium position of the drop in the trap causes significant, large-amplitude shape oscillations superimposed over the center-of-mass oscillations of the drop. An important observation here is that the shape oscillations due to the applied quadrupole fields result in sufficient deformations that act as triggers for the onset of the instability below the Rayleigh limit, thereby admitting a subcritical instability. The center-of-mass oscillations of the droplet within the trap, which follow the applied frequency, are out of phase with the applied AC signal. Thus the combined effect of shape deformations and dynamic position of the drop leads to an asymmetric breakup such that the Rayleigh fission occurs upward via the ejection of a jet at the north pole of the deformed drop.

Linear and non-linear stability analyses of Rayleigh-Bénard convection in water–copper and water–alloy nanoliquids

In this paper, we perform linear and non-linear stability analyses of Rayleigh-Bénard convection in a horizontal layer of water–copper and water–alloy nanoliquids. The corresponding eigen values for the problems involving the two nanoliquids are obtained and compared. The thermophysical properties of nanoliquids have been modelled as a function of the properties of water as base liquid, copper and alloy as nanoparticles. A non-linear analysis is performed using the energy method. The subcritical instability does not exist. As a limiting case the results of water are discussed with results of previous investigations, and a good agreement is found. The effect of nanoparticles is to destabilise the system. The results are depicted graphically.

Unsteady Aeroelastic Analysis of the Semi Aeroelastic Hinge Including Local Geometric Nonlinearities

A recent consideration in aircraft design is the use of folding wing tips, with the aim of enabling higher aspect ratio wings with less induced drag but also meeting airport gate limitations. Of particular interest is the concept of using in flight free-floating wing tips in order to reduce aircraft gust and maneuver loads. This study investigates the effects of local geometric nonlinearities on the dynamic stability, postflutter behavior and gust response of floating wing tips. A multibody formulation is introduced to account for finite rotations of rigid folding wing tips attached through hinges on a flexible airframe structure including also aerodynamic follower forces for the folding wing tip components. It is found that the postflutter behavior is characterized by super-critical limit cycle oscillations and no subcritical instability was observed for the analyzed cases. Moreover, the wing tip gust response can vary significantly when geometric nonlinearities are accounted for whereas a small impact was observed on the main airframe structure.

Study of Rayleigh–Bénard convection in a chemically reactive fluid using a generalized Lorenz model and the cubic–quintic Ginzburg–Landau equation

Analytical study of linear and weakly nonlinear stability analyses of Rayleigh–Bénard convection in a chemically reactive two-component fluid system is reported by considering physically realistic and idealistic boundaries. Analytical expression for the eigenvalue as functions of the chemical potential and the relaxation parameter is reported in the cases of stationary, oscillatory, and subcritical instabilities. The preferred stationary mode at onset is attributed to the assumption of a chemical reaction. The possibility of subcritical motion in a specific range of values of the relaxation parameter is reported and the threshold value of this parameter at which the transition from the subcritical to the critical motion takes place is documented for different values of the chemical reaction rate. The higher-order generalized Lorenz model leads to the reduced-order cubic–quintic, Ginzburg–Landau equation (GLE), and using its solution, the heat transport is quantified in steady and unsteady convective regimes. The drawback of the cubic GLE and the need for the cubic–quintic GLE for studying the heat transfer in the case of subcritical regime is explained. The possibility of having pitchfork and inverted bifurcations at various values of the relaxation parameter is highlighted. The steady Nusselt number plots clearly show that in the case of a supercritical bifurcation, a smooth transition takes place from unity in the conduction state to a higher value as the Rayleigh number exceeds its critical value predicted by linear theory. In the case of subcritical bifurcation, however, a discontinuous transition is observed.

The subcritical transition to turbulence of Faraday waves in miscible fluids

We study the development and the breaking process of standing waves at the interface between two miscible fluids of small density contrast. In our experiment, a subharmonic wave is generated by a time-periodic vertical acceleration via the Faraday instability. It is shown that its wavelength may be selected not only by the linear process predicted by the Floquet theory and favouring the most unstable modes allowed by the tank geometry, but also by a nonlinear mode competition mechanism giving the preference to subcritical modes. Subsequently, as the standing wave amplitude grows, a secondary destabilization process occurs at smaller scales and produces turbulent mixing at the nodes. We explain this phenomenon as a subcritical parametric resonance instability. Different approaches derived from local and global stability analysis are proposed to predict the critical wave steepness. These theories are then assessed against various numerical and experimental data varying the frequencies and amplitudes of the forcing acceleration.